People did use DAWs in the 90s, but in the early to mid 90s, the few DAWs that existed were generally only used by professionals. That is because the processing power of computers was not enough to handle the digital signal processing that even a basic DAW needs, and hard drive systems were not fast enough to stream audio in real time, so DAW systems required both software and dedicated hardware.
In 1991, Pro Tools was released, which was a DAW that could record and playback both MIDI and audio, with up to four simultaneous audio tracks. By 1994, dedicated "DSP farm" cards and SCSI RAID hard drive arrays brought the maximum possible track count up to 64. Pro Tools was the first widely popular DAW in professional studios, even though the very first digital hard disk recorder system was released in 1978. Also in 1991, Cubase first supported audio but it required TDM hardware (created by the makers of Pro Tools) for DSP.
Home studios didn't generally go digital until the late 90s. I myself used a tracker on the Amiga 3000UX in 1992 - 1993, just to make fun little tunes. In 1995, I took a class on MIDI sequencing based around Cakewalk, which at the time was only a MIDI sequencer. At that time, home studios mostly relied on 4 and 8 track analog cassette tape deck recording (like the Tascam Portastudio line). Other MIDI-only software in the late 80s and early 90s included MOTU Digital Performer.
I can't find the exact date, but definitely by 1998, Cakewalk software could record and mix several tracks of simultaneous audio without requiring dedicated DSP hardware (called "native"). I recall using Cakewalk as a mixdown destination for an analog 4-track recording sometime in 1996 or so. Also in the late 90s, Sonic Foundry's Sound Forge software, Bias Peak (released in 1996), Digital Performer, and Cubase (which added native audio processing in 1996) all could record, edit, and mix two or more tracks of audio on a regular home PC. Propellerhead Reason and Sonic Foundry (later Sony) Acid and Vegas were notable software releases at the end of the millennium, with Acid and Reason being the software that popularized looping, prior to Ableton releasing Live in 2001.
The first release of the Steinberg VST specification and SDK was in 1996, but the popularity of ASIO and the VST plugin format actually took a while to catch on. Some competing systems were proprietary, like MOTU's MAS system and Digidesign's (the makers of Pro Tools) Audio Suite and TDM systems. Other plugin formats were OS-based, like Microsoft's DirectX and Apple's Audio Units. As far as I know, DirectX is not used by DAWs today, but Apple Audio Units continues to be a major plugin format despite only being available on MacOS systems. My first DAWs in 1997 - 2002 were Cakewalk and Sony Vegas, which both relied heavily on the DirectX plugin format and were therefore Windows-only packages.
So the answer to your question is that DAWs did not even become available to the home user until the mid to late 90s, and it wasn't until the early 2000s that VST started to displace DirectX on Windows as the most popular plugin format. And on MacOS to this day, VST is not needed (although it is available). I myself use several DAWs for various purposes on a Mac and I never use the VST or ASIO standards for anything.
Also note that there were digital audio systems that were not computer based that had popularity in the 90s. For smaller studios and home studios, the Alesis ADAT format, released in 1992, allowed up to 64 tracks of digital audio to be recorded on several ADAT tapes, that were the same physical format as VHS video tapes. Jagged Little Pill by Alanis Morissette was recorded on ADAT.
The Otari RADAR dedicated multitrack hard disk recorder was first released in 1993, but was priced generally out of reach of the home studio crowd. Like ADAT, you couldn't mix or apply effects with a RADAR unit, it was just a digital version of a 24-track tape deck. So you still had to have an analog mixing console and any outboard effects to make a mix. From that point of view, it was a very expensive system, but being able to erase the hard disk and perform non-linear editing without having to cut tape or having tape wear out or having to calibrate tape heads was a big advantage.